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Extinction probabilities in predator–prey models

Published online by Cambridge University Press:  14 July 2016

S. E. Hitchcock
S. E. Hitchcock*
Affiliation:
Open University
*
∗Postal address: Faculty of Mathematics, Open University, Walton Hall, Milton Keynes, MK7 6AA, UK.
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Abstract

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

Keywords

IMBEDDED MARKOV CHAINAPPROXIMATION TO EXTINCTION PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 1 - 13
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research performed under the support of a grant from the Science and Engineering Research Council while the author was at Birkbeck College, University of London.

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